Properties

Label 2-2646-63.4-c1-0-33
Degree $2$
Conductor $2646$
Sign $-0.972 + 0.234i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 1.03·5-s − 0.999·8-s + (−0.517 + 0.896i)10-s + 0.267·11-s + (0.896 − 1.55i)13-s + (−0.5 + 0.866i)16-s + (3.41 − 5.91i)17-s + (2.19 + 3.79i)19-s + (0.517 + 0.896i)20-s + (0.133 − 0.232i)22-s − 5.46·23-s − 3.92·25-s + (−0.896 − 1.55i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.462·5-s − 0.353·8-s + (−0.163 + 0.283i)10-s + 0.0807·11-s + (0.248 − 0.430i)13-s + (−0.125 + 0.216i)16-s + (0.828 − 1.43i)17-s + (0.502 + 0.870i)19-s + (0.115 + 0.200i)20-s + (0.0285 − 0.0494i)22-s − 1.13·23-s − 0.785·25-s + (−0.175 − 0.304i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.972 + 0.234i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.972 + 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.176409415\)
\(L(\frac12)\) \(\approx\) \(1.176409415\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 1.03T + 5T^{2} \)
11 \( 1 - 0.267T + 11T^{2} \)
13 \( 1 + (-0.896 + 1.55i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.41 + 5.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.19 - 3.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.46T + 23T^{2} \)
29 \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.34 - 5.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.73 + 6.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.31 + 7.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.133 + 0.232i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.378 - 0.656i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.46 + 9.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.637 - 1.10i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.31 - 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.23 + 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.46T + 71T^{2} \)
73 \( 1 + (-2.70 + 4.69i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.46 - 7.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.29 + 5.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.07 + 15.7i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.523784199903799858639942501207, −7.73106472452346061399375813744, −7.12623884949234855364666883981, −5.85378360640521852647841145036, −5.45826548953075352606750129239, −4.33560031159880272351224622048, −3.63625533970583814828468624824, −2.82617742494612125003750554497, −1.65058224693382158802347277447, −0.34770590512463103051771986481, 1.43589135710035617476275694921, 2.83765577357234685836726861634, 3.86507072476603923433679103097, 4.35356984941006989398612551773, 5.47570514401010230782785587493, 6.12204452840794914544791791802, 6.86056791124238131244796747590, 7.85529017499870938658225706338, 8.110586353511309495048990972457, 9.092607234026790599097536890272

Graph of the $Z$-function along the critical line